Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure

Transcription

1 UHasselt Computational Mathematics Preprint eries Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure Hans van Duijn, Koondanibha Mitra and Iuliu orin Pop UHasselt Computational Mathematics Preprint Nr. UP-7-6 July 5, 27

2 Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure C.J. van Duijn,2, K. Mitra 3,4, and I.. Pop 4,5 Eindhoven University of Technology, Department of Mechanical Engineering 2 University of Utrecht, Department of Earth ciences 3 Eindhoven University of Technology, Department of Mathematics and Computer cience 4 Hasselt University, Faculty of cience 5 University of Bergen, Department of Mathematics Abstract The Richards equation is a mathematical model for the unsaturated flow through porous media. This paper considers an extension of the Richards equation, where non-equilibrium effects like hysteresis and dynamic capillarity are incorporated in the relationship that relates the water pressure and the saturation. The focus is on travelling wave solutions, for which the existence is investigated first for the model including hysteresis and subsequently for model including dynamic capillarity effect. In particular, such solutions may have non monotonic profiles, which are ruled out when considering standard, equilibrium type models, but have been observed experimentally. The paper ends with numerical experiments confirming the theoretical results. In this sense the original system of partial differential equations is solved by means of an implicit scheme. This relies on an equivalent, mixed formulation of the system. For solving the resulting nonlinear, time-discrete problems, a linear iterative scheme is proposed. Introduction Unsaturated flow through porous media is encountered in many applications of societal and engineering relevance. Examples in this sense are the groundwater flows, or the moisture dynamics in building materials. A commonly used mathematical model for such kind of processes is the Richards equation, which is obtained after inserting the Darcy law into the water mass balance equation. The two main unknowns in this equation are the water saturation (the percentage of the pore space in a representative elementary volume that is occupied by water) and the water pressure p. In standard porous media flow models, these two unknowns are related through the strictly decreasing capillary pressure function P c ( ), namely p = P c (), which is determined experimentally. Different types of functions and parameterizations are discussed e.g. in [3], the common assumption being that these dependencies are obtained under special, equilibrium conditions. More precisely, the experiments are carried out either for imbibition or for draining and not when these processes occur alternatively, and during an entire imbibition or drainage cycle each measurement has been done only after water stops redistributing inside the pores of the elementary volumes. uch models will therefore be called in what follows equilibrium type models. Corresponding author: k.mitra@tue.nl

3 In realistic applications, neither of these conditions are met. First, experiments reported e.g. in [5, 29] have revealed the hysteretic nature of the pressure - saturation relationship. More precisely, it was observed that the functions P c determined during infiltration and drainage are different. This motivated an extremely rich literature on mathematical models describing hysteresis. The play-type hysteresis model assumes a switch between imbibition and drainage capillary pressure-saturation curves whenever the saturation changes from increasing in time to decreasing in time or vice-versa. A mathematical formulation of this is given in [3], and the switch happens along vertical scanning curves. This poses nontrivial issues when analysing the resulting models and their numerical discretisations, which can be resolved by approximating the vertical scanning curves by monotone and non-vertical ones. In this sense, commonly used is the Lenhard-Parker model [33], where the scanning curves are rescaled versions of some predefined curves. A simplified version of it is proposed in [], where the scanning curves are oblique lines. Other hysteresis models build on concepts like percolating/nonpercolating phases [2, 24], or interfacial area based models [3, 35]. An overview of hysteresis models can be found in [5], whereas details on the numerical approximation of hysteresis in porous media models are given in [32]. In the present paper we consider the play-type hysteresis model for the pressure - saturation dependency but it is interesting to note that hysteresis can also be present in the relative permeability curve [34]. However, in the latter case this effect is less important in comparison to the former [5]. econd, when letting the water infiltrate in a homogeneous medium, experiments have revealed profiles that are conflicting with the profiles of the solutions to the equilibrium type models. For example, if the injection rates at the inflow are high enough, the obtained saturation profiles are non-monotone as the values at some locations inside the column are higher than at the inflow boundary (the so-called overshoot phenomenon, see [8]). In particular, the experiments in [7] show that although the saturation at some certain location is decreasing in time, the water pressure is non-monotone and exhibits a peak at moments when the saturation changes rapidly. This pleads for the inclusion of dynamic effects in the pressure-saturation relationship, as suggested in [22]. In mathematical terms, models like those mentioned above are evolution equations of pseudoparabolic type, or involve differential inclusions. uch models will be called below nonequilibrium type models. In this paper, we investigate how the solution profiles for unsaturated flow through a long, homogeneous porous column are affected by such non-equilibrium effects. The analysis is based on travelling waves (TW), allowing to reduce the model first to a nonlinear ordinary differential equation, and then to a dynamical system. This provides insight in the structure and behaviour of the solutions, and in particular how the non-equilibrium regime affects the profiles. The present analysis follows the ideas in [46], which studies the existence of TW solutions for reactive flow and transport models in porous media. In [7] TW solutions are analysed for nonlinear models that are similar to the Richards equation, but where higher order effects are included inspired by the ones describing dynamic capillarity. The nonlinear functions taken in [7] are of power-like type, in particular the flux function is convex. The existence of TW solutions is analysed, and in particular it is shown that oscillations behind the infiltration front may occur, depending on the magnitude of the dynamic effect. A similar analysis, but for two-phase flow models implying convex-concave flux functions is carried out in [44, 47, 48]. Also related are the diffusive-dispersive equations appearing as models for the phase transition dynamics, but in which the higher order terms are in terms of the spatial derivatives only [2, 6]. Though having a different motivation, the associated TW equation is similar to the one for the dynamic capillarity models, in particular since both involve a non-convex nonlinearity in the lower order terms. In this context, in [47] it is proved that the saturation profile may have overshoot in form of a plateau separated by two fronts (infiltration-drainage), similar to the ones 2

4 obtained in [8]. The dependence of the saturation value at such plateaus on the magnitude of the dynamic effect is proved rigorously in [47], and non-standard entropy conditions are defined for the shock solutions of the limiting hyperbolic case when the capillary effects are neglected. This analysis is extended to the case of degenerate models in [44, 48]. Due to the degeneracy in the model, the saturation remains between the physically relevant values, but the TW solutions may have discontinuous derivatives. The possibility of encountering non-monotonic TW profiles for various extensions of the Richards equation, including dynamic capillarity models, is evidenced numerically in [9, 2]. Finally, we mention [5] for a numerical study of the saturation and capillary pressure profiles for several of the hysteresis concepts discussed above, combined with dynamic capillarity. The present analysis consists of three parts. First the existence of TW solutions is analysed for the models involving hysteresis. The TW profiles are obtained by regularising the multi-valued function involved in the hysteretic term. In particular, we analyse the orbits associated with the TW solution in the saturation-pressure plane. We prove that in the initial and the final stages these orbits follow scanning curves that become vertical when the regularisation parameter vanishes, and in between they follow the corresponding primary curves (imbibition/drainage). Next, the case where dynamic effects are present in the pressure - saturation relationship is discussed. The existence of TW solutions is obtained and criteria ensuring their nonmonotonicity are provided. These include also situations where full-saturation is achieved. In the last part we discuss a numerical scheme for approximating the solution of the non-linear, pseudo-parabolic partial differential equations modelling the processes described above. The scheme is implicit, so at each time step one has to solve a nonlinear problem. In this context we propose an iterative method which is unconditionally convergent. Finally, numerical results validating the theoretical findings are provided. As will be seen below, the numerical solutions to the original model are reproducing nicely analytically predicted structures and properties of the TW solutions. 2 Mathematical formulation 2. Basic equations We consider the unsaturated water flow in a one-dimensional, homogeneous porous medium. Let t and x denote the time and space variable respectively. Assuming that the medium is vertical so that gravity effects are playing a role, a well accepted model for the flow is the Richards equation [3], φ t = [ κ k() ( )] p x µ x ρg. (2.) The unknowns in the model are the water saturation and the water pressure p. The relative permeability k( ) is a given, positive and increasing function that characterizes the medium and can be determined experimentally. The other quantities are parameters in the model and are assumed positive and known: µ and ρ are the the water viscosity and density, g is the gravitational acceleration, φ is the porosity of the medium, and κ its absolute permeability. The model is completed by a an equation describing the dependency between p and. For standard models, this dependency is algebraic, p = P c (), 3

5 where P c ( ) is a decreasing function. Its specific form is determined experimentally. As mentioned, the results available in the literature assume a local equilibrium and disregard the history of the system. Here we consider the non-equilibrium model proposed in [3], which combines dynamic effects in the p- relationship with a simple, play type hysteresis model. For a mathematical justification of the play-type hysteresis, based on the pore scale analysis, we refer to [43]. Let p imb ( ) and p drn ( ) be the primary imbibition and drainage capillary pressure curves [29] respectively. In the absence of the dynamic effects one has { p imb () for t > (infiltration), P c () = (2.2) p drn () for t < (drainage). Combining this with the vertical scanning curves the closure relationship can be written in the compact mathematical form ( p = P c, ) ( ) () P () sign, (2.3) t t where sign( ) is the multi-valued function (the signum graph) for u > sign(u) = [, ] for u = for u <. (2.4) The functions, P are defined as (also see Figure for an example in the dimensionless framework) = 2 (p drn + p imb ), and P = 2 (p drn p imb ). (2.5) To include the dynamic effects we refer to [22]. With τ being a damping parameter and f( ) a damping function (both non-negative), the model combining hysteretic and dynamic effects in the pressure-saturation relationship reads ( ) p () P () sign τf() t t. (2.6) In [22], a thermodynamic justification of such models has been given. Also, homogenisation techniques are employed in [8] for justifying the dynamic terms. For experimental studies concerning the value of τ and the shape of the function f we refer to [7]. 2.2 caling and assumptions In what follows we assume that water infiltrates in a porous column under both capillary and gravity effects. The column is assumed isotropic and homogeneous, implying that φ and κ are constants. We also assume that the column is insulated laterally, so the flow will be essentially one-dimensional, in the direction of the gravity. ince we consider here TW solutions, the column is assumed infinite. With σ being the air-water surface tension coefficient, we consider the reference quantities φ p = σ κ, L = p ρg, T = µφl ρgκ, τ = µl2 φ κ, (2.7) 4

6 and apply the rescaling x = x L, p = p p, t = t T τ = τ τ. (2.8) Observe that reference value for pressure is inspired by the J-Leverett relationship and the reference value for the damping parameter τ is consistent with [26]. Also, since the analysis below will involve infinite domains, we have first specified a reference pressure and based on it a reference length has been defined. Putting the scaled variables in (2.) and (2.6) and disregarding the to simplify the notation one obtains the dimensionless system t = [ ( )] p k() x x, (2.9) p () P () sign ( t ) τf() t. (2.) Next we state the assumptions on the nonlinear functions involved in the model. assumptions are justified from physical point of view. (A. ) The relative permeability k C ([, ]) is a continuously differentiable, increasing, bounded and convex function. There exists M k > s.t. for all [, ] one has k(), and k () M k ; The (A. 2) The damping parameter is positive, τ. satisfies f() > for all (, ). The damping function is continuous and (A. 3) The scaled primary drainage and imbibition pressure functions p imp, p drn : (, ] [, ) are continuously differentiable, strictly decreasing and satisfy p imb () = p drn () =, and p imb () < p drn () for all (, ); Pc.5 p imb p drn Figure : Dimensionless primary imbibition (p imb ) and drainage (p drn ) capillary pressure curves and their average ( ) as a function of saturation. The curves are based on the van Genuchten model [49] and the parameters are taken from experiments [52, p. 9]. An immediate consequence is that the functions P ± ( ) defined in (2.5) are in C ((, ]), satisfying P ± () = and P ± () > for all (, ). Also, for any δ >, a constant M P (δ) > exists s.t. M P (δ) < ( ) () < for all [δ, ]. Figure displays an example of primary drainage and imbibition curves and their average. To analyse the effect of hysteresis, which is modelled by means of a multivalued function, we consider a regularisation approach. With ε > being a small regularisation parameter, one can approximate the sign function by another function H ε : R R satisfying the following (A. 4) For each ε >, H ε is smooth, odd and strictly increasing. It satisfies H ε ( s) = H ε (s) and < H ε(s) H ε() = ε for all s R; 5

7 (A. 5) H ε is monotone w.r.t. ε: if ε > ε 2 > then H ε (s) < H ε2 (s) < for all s. Also, when ε, the functions H ε are approaching the signum graph: lim H ε(s) =, if s <, and lim H ε (s) =, if s >. (2.) ε ε Finally, H ε depends on ε in a continuously differentiable manner. Observe that H ε () = for all ε >, that H ε (s) decreases with ε for negative arguments s, and that this monotonicity is reversed for positive arguments. When sign is replaced by H ε in (2.), the regularised model for the pressure-saturation relationship becomes ( ) p = () P ()H ε τf() t t. (2.2) uch regularisation has been used in [39, 42] for proving the existence of weak solutions to such models, and for developing appropriate numerical schemes. It is an intriguing question whether this approach is motivated physically, and in particular how the parameter ε can be interpreted from physical point of view. In this sense one observes that in the play type hysteresis models, a switch from drainage to imbibition leads to a vertical jump between the corresponding primary curves in the pressure-saturation plane. In physical terms, the switch between the primary curves is made along vertical scanning curves. When considering regularised models, these scanning curves have a steep but finite gradient. This is actually observed in scanning curves obtained from experiments in [29]. As will be seen below, the same profiles are predicted by the TW analysis, and the scanning curves become steeper when ε is decreased. Another motivation for considering regularised models can be found in [5], where the play type hysteresis is viewed as a friction-controlled backslash process. This means that dissipative forces, which are mostly continuous in porous media, are responsible for it. At the pore scale, hysteresis occurs because of the difference in the advancing and receding contact angles of the wetting phase, which is a continuous phenomenon and hence jump phenomena should not be expected. Based on the above, ε can be seen as a physical parameter, or at least can be used to fit more realistic P c - scanning curves. Having this in mind, in the subsequent discussions we will analyse first the case ε > and then the limiting case of ε. Before doing so we mention that (2.9), combined with the constitutive relationship (2.) or its regularised counterpart (2.2), becomes a nonlinear, pseudo-parabolic equation. In general, one cannot expect that solutions exist in a classical sense. We refer to [5, 6, 2, 3, 4, 25, 28, 27, 4, 42, 43] for results concerning the existence and uniqueness of weak solutions for hysteresis models, dynamic capillarity models, or for models including both effects. In particular we refer to [4, 42, 3] where, as suggested in [3, 4], (2.2) is used to express t as a function of and p. We rely on the same idea for the TW analysis below. 2.3 Travelling wave formulation To simplify the analysis and to understand the profile of the solutions to the regularised mathematical model (2.9), (2.2) we look for TW solutions. We assume that the solutions have profiles that do not change in time, but travel with a velocity c that will be determined later. pecifically, we extend the domain (the porous medium) to the entire real axis R and assume that the saturation and the pressure depend on the TW variable ζ = ct x. Note that this choice is the opposite of x ct, which is commonly used in the TW analysis, but it is convenient 6

8 for the analysis below. For the ease of presentation, we define the negative pressure u = p, and assume that and p only depend on ζ, (x, t) = (ζ) and p(x, t) = u(ζ), with ζ = ct x. (2.3) The wave velocity c R will be determined below. With the travelling wave transformation (2.9) and (2.2) become c = (k()(u + )), (2.4) u = () P ()H ε (c ) cτf(), (2.5) defined for all ζ R. In the above equations denotes the differentiation w.r.t. ζ. Furthermore, we assume that the saturation and the pressure admit horizontal asymptotes at ±, lim (ζ) =, ζ lim u(ζ) = p B, ζ lim (ζ) = T, (2.6) ζ lim u(ζ) = p T. (2.7) ζ for given saturations T, satisfying T > > and pressures p T and p B. Observe that since u = p, p T and p B are the additive inverses of the actual pressure. After integrating (2.4), we use the behaviour of and u at ± to deduce that Moreover, from (2.5) one obtains that lim ζ ± u (ζ) = lim ζ ± (ζ) =. lim u(ζ) = ( ) and lim u(ζ) = ( T ), ζ ζ which provides a necessary condition for the existence of TW solutions. We have Proposition 2. A necessary condition for the existence of TW solutions is that the components in the left and right states are compatible, namely p α = ( α ) (α {T, B}). In what follows, this compatibility condition is assumed unless stated otherwise. Integrating (2.4) and using the boundary conditions to determine the wave speed and the constant of integration gives u = G(;, T ) and c = k( T ) k( ) T. (2.8) The function G : (, ] R depends on the parameters and T satisfying < < T <, and is defined as G(;, T ) := c( ) + k( ) k() = c( T ) + k( T ). (2.9) k() The last equality follows from the definition of the wave speed c. In the subsequent analysis we investigate the TW solutions for hysteresis (τ = ) and dynamic capillarity cases (p imb () = p drn () for < implying P () = ) separately. Recalling (2.6)-(2.7) and Proposition 2., the points E α = ( α, p α ) with p α = ( α ) (α {T, B}) and T > are equilibria for the dynamical system (2.4)-(2.5). Therefore, to find and analyse TW solutions of the non-equilibrium model (2.9), (2.2) we seek orbits 7

9 connecting the points E B and E T in the -u plane. Clearly, corresponding waves are unique up to a translation in the ζ coordinate. To fix the orbit we also assume () = 2 ( + T ) and (ζ) < 2 ( + T ) for all ζ <. (2.2) We will see later, the inequality in (2.2) is needed as will become non-monotonic if the equilibrium point E T becomes a stable spiral sink. From now on while discussing travelling waves or orbits we implicitly assume that (2.2) is satisfied. 3 Capillary hysteresis We start the analysis by considering first only hysteretic effects in the absence of dynamic ones (τ = ) in the capillary pressure. ince H ε is invertible, one can define the function Φ ε : (, ) R, Φ ε (r) = c H cε (r), (3.) where c is the wave speed in (2.8). The particular choice of Hcε instead of Hε simplify the writing below. Recalling the Assumptions (A.4) and (A.5) one gets is made to Proposition 3. Φ ε is a smooth, odd and increasing function satisfying Φ ε() = ε for all ε >. Also, given two regularisation parameters ε,2 s.t. ε 2 > ε > one has Φ ε (r) < Φ ε2 (r) for all r (, ). Finally, lim ε Φ ε (r) = for all r (, ). 5-5 Φ ε Φ ε,ε =.5 Φ ε2,ε 2 =.5 - Figure 2: The graph of Φ ε. The plots are for Φ ε (r) = εr r 2 and two ε values are ε =.5 and ε 2 =.5. r The proof is straightforward. Figure 2 shows the graph of Φ for different values of ε. Putting τ = in (2.5) and rearranging the resultant equation we have the following system ( P + ) () u = Φ ε P (3.2) () u = G(;, T ). (3.3) For the TW analysis we consider the cases T = and T < separately. 3. The case T = Below we will use the following inequality, which follows directly from Assumption (A.5) and (2.5) < () u P < for all < <. (3.4) () The main result of this section is Theorem 3. Let < < T = and E B = (, ( )), E T = (, ). (a) Let ε > be fixed. The system (3.2)-(3.3) has a unique orbit ( ε, u ε ) connecting the points E B and E T. This orbit is increasing in and decreasing in u. Consequently, for any [, ] there exists a unique ζ ε R s.t. ε (ζ ε ) =, and similar result holds for u [, ( )]. 8

10 (b) The orbits ( ε, u ε ) are well ordered with respect to ε and do not intersect except at the equilibrium points E B and E T. pecifically, if ε 2 > ε > and ε (ζ ) = ε2 (ζ 2 ) = for some (, ) and ζ,2 R, then u ε2 (ζ ) > u ε (ζ 2 ). (c) Let (, ] be fixed. For arbitrary ε >, let ζ ε, R be s.t. ε (ζ ε, ) = and w ε, = u ε (ζ ε, ). Then lim ε w ε, = p imb (). Observe that for any ε > Theorem 3. provides the existence of TW solutions to the regularised system (2.9), (2.2) that connect the states (, ( )) and ( T, ( T )) = (, ). Here uniqueness is stated under the assumption in (2.2). Thus once the orbit is known, one can let ε to obtain the existence of TW solutions for the play-type hysteresis model. Further, due to the monotonic behaviour of the orbits, the TW solutions are monotone in both components and in particular no overshoot is possible in either pressure or saturation. Finally, once the monotonicity is proved, for any fixed ε >, the functions ε : R (, ) and u ε : R (, ( )) are one to one. This is being used in the last point of Theorem 3. when, for arbitrary (, ), a TW argument ζ ε, and the pressure w ε, were defined. Moreover, for any ζ < ζ ε, one has ε (ζ) < and u ε (ζ) > w ε,. imilarly for ζ > ζ ε, one has ε (ζ) > and u ε (ζ) < w ε,. Recalling T = this justifies Definition 3. Let ε > and ( ε, u ε ) be the orbit of (3.2)-(3.3) connecting the equilibria E B and E T. Let (, ) and ζ ε, R be such that ε (ζ ε, ) =. The functions η ε, w ε : (, ) R are defined as η ε () = ζ ε,, w ε () = u ε (ζ ε, ). Note that the definition is given for (, ), but it can be extended to = and =, with w ε taking the values ( ), and respectively, and η ε becoming. Further, since the definition makes sense whenever is monotonic, if this does not hold globally, the functions η ε and w ε can still be defined but restricted to intervals where the saturation is monotone. Finally, from (3.2) and (3.3) one obtains w ε() = dw ε d () = G(;, ) ( ). (3.5) () w ε Φ ε P () This will be used in the analysis below. To prove Theorem 3. we need some intermediate results. We start with Proposition 3.2 The region H = {(, u) : and p imb () u ()} is positive invariant for the dynamical system (3.2)-(3.3). Proof ince k( ) is a convex function it follows that G(;, ) for any [, ]. Also ε whenever p imb () u (). Therefore any orbit ( ε, u ε ) will be monotone in both components as long as it remains in H, and the function w ε introduced above is well defined. 9

11 u E B H p imb p drn T = Figure 3: The invariant set H in the -u plane. The arrows indicate direction of orbits with ζ increasing. Referring to Figure 3, since ε = and u ε < along the graph of, the orbit cannot leave H through the upper boundary. The same holds for the vertical boundary =, since along it one has ε >. Finally, as the orbit approaches the primary imbibition curve p imb one has ε + and therefore dw ε. ince d p imb <, this implies that the orbit cannot leave H through the lower boundary as well. Hence H is invariant. The next proposition is characterising the equilibrium point E B. Proposition 3.3 E B is a saddle type equilibrium. Proof Linearizing (3.2)-(3.3) around any equilibrium point E α = ( α, ( α )) (α {B, T }) yields the characteristic equation λ 2 Φ ε() ( α ) P ( α ) λ + Φ ε() (k ( α ) c) k( α )P =. (3.6) ( α ) ince k is convex one has k ( ) < c < k (). For E B, by Proposition 3. and Assumption (A.3), the last term on the right is negative, which proves the proposition. Remark 3. ince Φ ε() = ε, one can immediately determine the asymptotic behaviour of the positive eigenvalue in (3.6) for the equilibrium point E B. pecifically, with C,2 > properly chosen one has λ +,B,ε = C 2 ε 2 + C 2 ε C ε = O( ε), as ε. Now we can prove Theorem 3.. Proof (a) With given ε >, we consider the unique orbit ( ε, u ε ) of the dynamical system (3.2)-(3.3) that leaves the saddle E B along the unstable direction corresponding to λ +,B,ε in Remark 3., and for increasing ε. By direct calculation one obtains that the corresponding eigenvector points into the region H. By Proposition 3.2, the orbit ( ε, u ε ) remains in H and therefore it is monotone in both components. Furthermore, as ζ, since the orbit does not leave H and its monotonicity prevents it from returning to E B or approaching a limit cycle, the orbit approaches the second equilibrium E T = (, ). In other words, we have proved the first point in Theorem 3.. (b) For the second point we observe that the monotone behaviour of the orbit ( ε, u ε ) allows using the function w ε introduced in Definition 3.. Letting in (3.5) gives w ε( ) = ( ) 2 ( ) + + 4(c k ( ))P ( ) k( )Φ ε()( ( )) 2. (3.7)

12 Let now ε 2 > ε >. ince Φ ε() = ε, from (3.7) one obtains w ε 2 ( ) > w ε ( ). As w ε2 ( ) = w ε ( ) = ( ), this implies that a δ > exists s.t. w ε2 () > w ε () for all (, + δ). Assume that the two orbits intersect in the interior of H, i.e. a (, ) exists s.t. w ε2 ( ) = w ε ( ) = u. Then, for the left limits of the derivatives one has w ε ( ) w ε 2 ( ). This gives a contradiction since at the point (, u ) H it holds that w ε 2 ( ) = G( ;, ) ( ( ) u ) > Φ ε2 P ( ) G( ;, ) ( ( ) u ) = w ε ( ). Φ ε P ( ) (c) From the above we know that for any (, ), w ε () decreases with ε and is bounded from below by p imb (). This implies that along any sequence ε the sequence w ε () has a limit. This allows defining the function w : [, ] [, ( )], w() = lim ε w ε (). Clearly, w() p imb () for all, and w is decreasing. Now suppose that a (, ) exists s.t. w( ) > p imb ( ). Then, with δ > small enough, δ > and w ε () > w() > p imb ( δ) for all ( δ, ) and ε >. Observe that all orbits are passing through the region (see Figure 4) R = {(, u) : δ/2 < < and p imb ( δ) < u < ()}. (3.8) Observe that on [ δ 2, ] the function G is bounded from below away from, G C for some C >. Defining ( ) () u m = P, () sup (,u) R we have < m < since R does not touch the curve p imb. monotonicity of Φ ε, in R we have Therefore by mean value theorem we get w ε() ( ) P imb ( ) w ε ( δ) w ε ( ) From (3.5) and due to the C Φ ε (m). (3.9) Cδ Φ ε (m). While the left hand side is bounded, the right hand side goes to as ε, leading to a contradiction. Therefore lim w ε () = p imb () and the convergence is uniform on any closed ε interval [ + µ, ] with µ >.

13 u E B δ H T = R p imb p drn ( ε, u ε ) Figure 4: The saturation and the region R for ε >. Passing ε gives TW solutions to the play-type hysteresis model. These are the limit of the orbits obtained in the regularised case, satisfying (2.2). We have Corollary 3. Let ζ ε R be s.t. u ε (ζ) = p imb ( ) and let ε = ε (ζ ε ). Then lim ε ε = and lim ζε =. ε Before giving the proof we observe that in view of the convergence result in Theorem 3. (c), this corollary shows that for ε the orbits are becoming vertical when approaching E B. Proof ince the orbits ( ε, u ε ) are ordered, ε decreases with ε. Moreover, ε since w ε ( ) = ( ), so there exists = lim ε along any sequence ε. With this, the first ε result is obtained by contradiction, repeating the proof for point (c) of Theorem 3.. To show that ζε as ε we observe that the function η ε in Definition 3. is the inverse of ε. By (3.2), it satisfies η ε = Φ ε ( () u P () ). (3.) As ε = (ζ ε ) and ε () = ( + T )/2, integrating (3.) and using (3.5) gives ζ ε = ε() ε d ε() w = ε() Φ ε ε G(;, ) d. ince ε, for any δ > there exists a µ = µ(δ) s.t. < ε < +δ for all < ε < µ(δ). As G C ([, ]]), with m = max{ G (), [, 2 ( T + )]} it holds ζ ε m ε() ε w ε() d 2 ( T + ) w ε() d =: m +δ m h ε. (3.) To evaluate h ε, the last integral in the above, we use the uniform convergence of w ε to p imb. However, in h ε the derivative w ε appears, therefore we proceed as follows h ε = w ε( + δ) δ 2w ε( 2 ( T + )) ( T ) 2 ( T + ) w ε () +δ ( ) 2 d. The uniform convergence of w ε to p imb proved in Theorem 3. gives lim h ε = h = p imb( + δ) ε δ = 2 ( T + ) +δ 2p imb( 2 ( T + )) T p imb ( ) d, 2 ( T + ) p imb () +δ ( ) 2 d as p imb C. Therefore a µ (ν) µ exists s.t. h ε > h ν for all ε (, µ (ν)). 2

14 Using this and (3.), and observing that Assumption (A.3) guarantees the existence of a m p > s.t. m p p imb () for, if < ε < µ (ν) one has ζ ε m 2 ( T + ) +δ p imb ( ) d ν m m p m ln ( T 2δ ) ν m. (3.2) As δ was taken arbitrarily small, from this we can conclude that ζ ε as ε. 3.2 The case T < We consider now the case when the top and bottom saturations satisfy < < T <. In the analysis we use the region and its subregions (see Figure 5) We first analyse the case when ε > Properties for fixed ε > H = {(, u) :, p imb () u p drn ()}, H = {(, u) : T, P imb () u ()}, H 2 = {(, u) : T, P imb () u ()}, H 3 = {(, u) : T, () u P drn ()}, H 4 = {(, u) : T, () u P drn ()}. The key properties of the orbits are stated in Theorem 3.2 Let < < T < and ε >. Then the following holds (a) There exists a unique orbit ( ε, u ε ) satisfying (3.2),(3.3),(2.2) and connecting E B and E T. (b) There exists a ε m > s.t. E T is a stable spiral sink whenever < ε < ε m. u E B H 4 E T H T H 3 H2 p imb p drn u E B H 4 E H 3 T H H 2 p imb p drn T Figure 5: The regions H, H 2, H 3 and H 4 for the cases T < (left) and T > (right) where is defined as p drn ( ) = ( ). The arrows indicate direction of orbits with ζ increasing. 3

15 Proof (a) We observe that, repeating the proof for the T = case, the equilibrium E B is a saddle and all orbits leaving E B along the unstable direction and for increasing ε enter the region H introduced in Proposition 3.2. Also, no orbit can leave the region H defined above through the primary curves p imb () and p drn (). Now we let be s.t. p drn ( ) = ( ). Then two cases can be identified, T < and T. The case T < : As seen from the left picture in Figure 5, the orbit leaving E B, for increasing ε, enters first the region H. Then there are four possibilities (see Figure 5). The orbit goes through H 2, H 3, H 4 and returns to E B. 2. The orbit goes through H 2, H 3, H 4 and leaves H 4 through the segment (, p B ), (, p drn ( )). 3. The orbit goes through H 2, H 3, H 4 and then leaves H 4 through the arc (, ()) between E B and E T. This in turn gives rise to two possibilities: A. The orbit moves around E T but does not approach it. B. The orbit ends up in E T. The case T : In this case, if the orbit enters from H 3 to H 4 at some ζ = ζ 3-4, u ε (ζ 3-4) < p drn ( T ) < p drn ( ) = p B. But in H 4, u ε is decreasing, hence u ε < p B for all arguments ζ > ζ 3-4, which rules out the first two possibilities (possibility and 2) above. To show that actually 3.B is the only possibility in both cases, we follow an argument from [7], based on Gauß Divergence Theorem. We define the vector-valued function F : H R 2, F(, u) = (Φ ε (, u), G()), (3.3) and denote its components by F and F u respectively. A direct calculation gives F = Φ ε (P ) 2 ( P P + up ), where the arguments and u are disregarded. Hence, for (, u) H one has Φ ε (P F(, u) < ) 2 ( P P + P p imb ) if P () < Φ ε (P ) 2 ( P P + P p drn ) if P () > The last factor in the first inequality gives 4 (p drn + p imb ) (p drn p imb ) 4 (p drn + p imb )(p drn p imb ) + p imb 2 (p drn p imb ) = 4 (2p drnp imb 2p imbp imb ) = 2 (p drn p imb )p imb < imilarly, in the second inequality one gets 2 (p drn p imb )p drn <. Thus we have shown that F(, u) < for all (, u) H. (3.4) We can now investigate the possibilities mentioned above. To rule out the first two possibilities in the case T < we define the domain Ω bounded by the closed orbit, or by the orbit and the segment (, p B ), (, p drn ( )) (see Figure 6). Let the orbit intersect the segment (, p B ), (, p drn ( )) at the point T. o for possibility, T is simply E T. By (3.4) one has T EB EB > F = F ˆn + F ˆn = F, Ω E B T T 4

16 u H 4 Ω E B E T H H 3 H2 p imb p drn ( ε, u ε ) u T H 4 Ω E B E T H H 3 H2 p imb p drn ( ε, u ε ) T T Figure 6: Possibility (left) : orbit returning to E B after going through regions H, H 2, H 3 and H 4. Possibility 2 (right) : orbit exiting region H 4 through the segment (, p B ), (, p drn ( )). with the last integral on the right appearing only in the second possibility listed above. ince F in the region H\H, this gives a contradiction. Finally, to eliminate 3.A we observe that by the Poincaré-Bendixson Theorem, if the orbit does not end up in E T then it must approach a limit cycle around E T. However, one can use again the argument above, to show that limit cycles do not exist. o, the only possible behaviour of the orbits is as stated in possibility 3.B. This is displayed in the left plot of Figure 7. (b) Having proved the existence of an orbit connecting E B and E T, showing that the orbit forms a stable spiral around E T for small enough ε is a matter of calculation. Using the properties of Φ ε, and the convexity of k T in (3.6) it is easy to show that for small values of ε, the eigenvalues corresponding to equilibrium point E T will be complex with negative real part. This completes the proof. The left plot in Figure 7 shows the phase portrait in the -u plane. In the right plot one has orbit component as function of ζ, in the case when E T is a stable spiral. Note the usage of ζ = x ct instead of ζ, which is because in the original problem (with x and t as independent variables) the left state (x = ) corresponds to T and right state (x = ) corresponds to. This convention will be useful when comparing with numerical solutions to (2.9)-(2.2). u E B p imb p drn ( ε, u ε ) R ε T 2 ( + T ) ε ( ζ) L ε R ε ζ T Figure 7: (left) Orbit connecting the saddle point to the spiral sink E T, and (right) the profile of as a function of ζ = x ct. 5

17 3.2.2 Properties for the limit case ε Knowing now the structure of the orbits for fixed ε >, we study the limit behaviour ε. In certain aspects, the results obtained for T = and for T < are quite similar. The major difference is in the fact that the orbits are not monotone anymore. In consequence, the function w ε introduced in Definition 3. can only be defined as long as ε remains monotone. Clearly, when starting from E B the monotonicity is lost for the first argument ζ where ε (ζ) = T. We define ζε T as ζε T = min{ζ R/ ε (ζ) = T }. (3.5) From now on we refer to the function w ε as the one obtained for ζ (, ζ T ε ]. With this one has Proposition 3.4 (a) For any (, T ), w ε () p imb () as ε. The convergence of w ε to p imb is uniform on [ + δ, T ] for any δ >. (b) As long as T the orbits ( ε, u ε ) are well ordered w.r.t ε >, and do not intersect. The proof is absolutely similar to the one for Theorem 3. and is skipped here. For the case T =, Corollary 3. is stating the limit behaviour of the orbits when ε. The type of the equilibrium E B remains unchanged in the case T < and a similar result can be proved in this case too. pecifically, if ζε R is s.t. u ε (ζ) = p imb ( ), for ε = ε (ζε ) one has lim ε ε = and lim ζε =, ε and the corresponding orbits become vertical when approaching E B. The situation changes for E T since the orbits ( ε, u ε ) become stable spirals. To understand this behaviour we let ζ ε = min{ζ R/ ε (ζ) = T } and define R ε = sup{ ε (ζ) : u ε (ζ) = ( ε (ζ))} and L ε = inf{ ε (ζ) : u ε (ζ) = ( ε (ζ)), ζ > ζ ε }, and prove the following Proposition 3.5 For ζ ε, L ε and R ε introduced above, one has lim ζ ε =, ε lim L ε = T and lim R ε = T. ε ε Proof The proof for ζ ε is almost identical to the proof of Corollary 3.. For the remaining part we only consider R ε, the proof for L ε being similar. Clearly, R ε T. Assuming that a δ > and a sequence ε k exists such that R εk > T + δ for all k N. Let { R = (, u) : T + δ } 2 < < T + δ and p imb ( T ) u (). Clearly, all orbits pass through R. Letting M = sup (,u) R ) ( ) () u P () ( one has M < and Φ () u ε < Φ P () ε (M) for all (, u) R. From (3.5) and recalling that k is convex, for any ( T + δ 2, < T + δ ) one has w ε k () > G(;, T ) Φ εk (M) > (k() k( T )) c( T ). k( T + δ) Φ εk (M) 6

18 Integrating the above over ( T + δ 2, T + δ ) and using the properties of k, a constant C > depending on δ but not on ε exists s.t. ( w εk ( T + δ) w εk T + δ ) > C 2 Φ εk (M). In the above, the difference on the left is bounded by ( T ) p imb ( T + δ). However, by Proposition 3., the ratio on the right goes to when ε, which gives a contradiction. This implies that R ε T for ε. u E B lim ζ ε( T ) ε = lim ζ ε( ) ε = p imb p drn lim ε ( ε, u ε ) E T T p B p imb ( ) T p T p imb ( T ) ( ζ) u( ζ) ζ Figure 8: Orbit for limiting case ε in -u plane (left); and saturation and pressure profiles for the limiting orbit as a function of ζ = x ct (right). Remark 3.2 Propositions 3.4 and 3.5 characterize the behaviour of the orbits in the limiting case ε. They are approaching vertical segments at = and = T, and in between the primary imbibition curve (see Figure 8). Possible oscillations can appear around E T when T <. As ε, these oscillations are damped in the component, but we are unable to show similar behaviour for the pressure. Computational results shown in Figures?? indicate that pressure oscillations do not decay as ε decreases. However, these oscillations cannot be observed in reality for ε as they are pushed towards infinity. Proceeding as in Corollary 3., one can show that ζε as ε and a similar result holds for the other side, determined by T. In another words, the oscillations move to and at any finite point the limiting waves are monotone in both saturation and pressure and they lie on the primary imbibition curve. 4 Dynamic capillarity Now we discuss the case without hysteresis, but include dynamic effects in the P c - relationship. More precisely, we assume that the primary curves in (2.5) are the same, p imb = p drn, giving P () = and () = p imb () for all. For the ease of presentation, as many results in this case are similar to the ones for the hysteresis model, we still use the notations P ±. At the same time we now take τ > and thus (2.) and (2.2) become u = () τf() t. With the TW velocity c given in (2.8), the dynamical system (2.4)-(2.5) associated to the TW solutions become, = () u, (4.) cτf() u = G(;, T ). (4.2) 7

19 As before, we seek orbits that connect the equilibria E B = (, ( )) and E T = ( T, ( T )), where < < T. To fix the ideas we normalize the orbits by assuming that () = ( + T )/2. We mention that the analysis here completes the ones in [7, 44, 47, 48]. In the following analysis we distinguish the cases, f L (, ) and f / L (, ). In the former case one can define the primitive function F : [, ] [, ), F () = f(ϱ)dϱ. By Assumption (A.3) F is one to one when its range is restricted to [, F ()]. With this, the transformation Y = F (), impying = F (Y ) (4.3) is well defined and (4.)-(4.2) becomes Y = ((Y )) u cτ (4.4) u = G((Y );, T ). (4.5) Observe that this system is similar to the one obtained for a constant damping function, f, as the functions (( )) and G(( )) have the same general properties as ( ) and G( ). Hence we start analysing the existence of TW solutions and their properties by replacing (4.) with the simpler equation = () u, (4.6) cτ as the analysis for the system (4.6), (4.2) can be immediately transferred to the general case when f is L. Moreover, the same applies for the case when f L, as this is different from f L only if T =. However, the case f L gives a natural framework in which the saturation remains within the physically relevant range, [, ]. 4. General behaviour of the orbits As for the hysteresis case, in this part we analyse the existence of orbits of the system (4.6), (4.2) connecting the equilibrium points E B and E T. Clearly, these orbits will depend on τ, motivating the notation ( τ, u τ ). Below we use the regions H = {(, u) : T, u ()}, H 3 = {(, u) : T, () u}, u E B H 4 H 3 E T H H 2 T Figure 9: The directions followed by the orbits in the -u plane for dynamic capillary case. H 2 = {(, u) : T, u ()}, H 4 = {(, u) : T, () u}. Figure 9 shows the directions followed by the orbits of the system (4.6), (4.2). Note that if an orbit goes through H, there it is monotone in both components, namely u τ < and τ >. Hence an orbit can only exit H through the line = T. A straightforward calculation shows that the eigenvalues for the linearization of (4.6), (4.2) around E α = ( α, ( α )) (α {B, T }) are λ ± τ = P + ( α ) 2cτ ( ) ± 4cτ(k ( α ) c) k( α )( ( α )) 2. (4.7) 8

20 ince k is convex, one has k ( T ) > c > k ( ), which shows that E B is a saddle point. Further, the unstable orbit leaving E B to the right enters the region H. To understand its behaviour as ζ we begin with Proposition 4. Given τ >, the orbit ( τ, u τ ) leaving E B into H either approaches E T from H as ζ, or leaves H through vertical line = T. Proof In view of the monotonicity inside H, if ( τ, u τ ) does not leave H through its right boundary it will approach an equilibrium contained in H and at the right of E B. ince k is a convex function, the only such point is E T. As for the hysteresis model, all orbits ( τ, u τ ) are monotone between (, T ). o, similar to Definition 3., with the stated normalization τ () = ( + T )/2 it is possible to define the functions η τ, w τ : (, T ) R for the dynamic capillarity model as well. More precisely, for any (, T ), a unique ζ w exists s.t. τ (ζ w ) = and τ (ζ) < for all ζ < ζ w. With this, w τ () = u τ (ζ w ) and η τ () = ζ w. Also one can extend w τ to the closed interval [, T ]. We emphasize on the fact that the functions are defined as long as τ remains increasing. In particular, this holds until the orbit leaves H H 2. imilar to (3.5), w τ satisfies the differential equation w τ () = τcg(;, T ) () w τ. (4.8) The propositions below explain how the orbits ( τ, u τ ) depend on τ, before they leave H. Proposition 4.2 For the family of functions w τ introduced above one has (a) w τ uniformly in [, T ] as τ. (b) For any (, T ], w τ () as τ. Proof We define the family of functions v τ : [, T ] [, ), v τ () = () w τ (). Note that since ( τ, u τ ) H, v τ is always positive. By (4.8) we get 2 (v2 ) () = vv () = cτg(;, T ) + v( ) cτg(;, T ). (4.9) Integration from = to an arbitrary (, T ) gives with K = c T v 2 () 2cτ G(ϱ;, T )dϱ 2cτ G(ϱ)dϱ. This implies () w τ () T G(ϱ;, T )dϱ = 2τ K, 2τ K. (4.) Observing that K does not depend on, the conclusion follows immediately. For the second part, assume there exists L > and (, T ] s.t. w τk ( ) > ( ) L for a sequence {τ k } k N going to infinity. ince w τk is strictly decreasing in [, T ] we have () w τk () < () ( ) + L if < <. ince G(;, T ) < in H integration of (4.8) gives w τk ( ) = w τk ( ) + cτ k < ( ) + cτ k G(ϱ) (ϱ) w τk (ϱ) dϱ G(ϱ;, T ) (ϱ) ( ) + L dϱ = ( ) cτ k K s, (4.) 9

21 with K s = G(ϱ;, T ) (ϱ) ( )+L dϱ. Clearly, K s >. ince lim k τ k =, this contradicts the assumed boundedness of w τk and the proposition is proved. The orbits depend continuously and monotonically on τ, as follows from Proposition 4.3 For all [, T ], w τ () is continuously decreasing w.r.t τ. Proof The proof for the monotonicity follows the arguments in the proof of Theorem 3., point (b) and is omitted. For the continuity we take (, T ] and < τ < τ 2, and use again the functions v ı = w τı, ı {, 2}. From (4.9) and using the monotonicity of w τ w.r.t. τ one obtains 2 (v2 2 v 2 ) () = c(τ 2 τ )G(;, T ) + (v 2 v )()( ) () < c(τ 2 τ )G(;, T ). With K defined above, integration gives < v 2 2() v 2 () < 2(τ 2 τ ) K, which implies the continuity w.r.t. τ of v and consequently of w τ. u E B < τ < τ 2 < τ 3 H 4 E T H H 2 T ( τ, u τ ) ( τ2, u τ2 ) ( τ3, u τ3 ) H 3 Figure : The dependency of the orbits ( τ, u τ ) on τ for < < T. From the discussion so far we conclude that the orbits ( τ, u τ ) are close to the graph of for small values of τ, but move away from it as τ increases, and for (, T ]. This situation is presented in Figure. In the remaining part of this subsection we focus on the behaviour of the system beyond the point = T. The main goal is to show that orbits connecting E B and E T exist for all values of τ >. In Theorem 4. we show this for small values of τ and for larger τ values the existence is shown in Theorems 4.2 and 4.3. Theorem 4. Let {( τ, u τ )} τ> be the family of orbits of (4.6), (4.2), originating from E B and entering H. Then there exists a τ > s.t. w τ ( T ) =. For all τ (, τ ] the system (4.6), (4.2) has a unique orbit ( τ, u τ ) satisfying τ () = ( + T )/2 and connecting E B and E T. Proof The existence of a τ for which w τ ( T ) = follows directly from Propositions 4.2 and 4.3. Also, w τ ( T ) < for τ > τ and w τ ( T ) > for τ < τ. u E B H 4 ζ Z ζ R E T H 2 H ζ T T ( τ, u τ ) ζ P H 3 Figure : Behaviour of the orbit ( τ, u τ ) for τ τ. 2 To understand the behaviour of ( τ, u τ ) for τ < τ we recall Proposition 4., which states that the orbit either approaches E T or enters H 2 through = T at a finite ζ T. In the latter case, which is displayed in Figure, u τ becomes increasing for ζ > ζ T. With τ < τ, since () = < u τ (ζ T ) < ( T ) the orbit must intersect the graph of at some ζ = ζ P and enter H 3, where τ becomes decreasing whereas u τ is still increasing. We claim that the orbit either approaches E T, or enters H 4 for some ζ = ζ 3.

22 To see this, assume that a δ > exists s.t. τ (ζ) T + δ for all ζ > ζ P. As τ is bounded and decreasing, the limit lim ζ τ (ζ) exists and is finite. Denoting it by τ we have τ T + δ. Further, since u τ is only bounded from below, a similar reasoning shows that either lim ζ u τ (ζ) = ũ τ [ ( τ ), ) or u τ. ince τ is decreasing with ζ and bounded from below, lim ζ τ =. From (4.6) one gets ũ τ = ( τ ). Therefore u τ has a (finite) limit as ζ and from (4.2) we get lim ζ u τ =. In other words, ( τ, ũ τ ) is an equilibrium point, which is not possible since k is a convex function and therefore G has only two zeros. This rules out the possibility that τ is bounded away from T, so either lim ζ τ (ζ) = T, or the orbit enters H 4 at some finite argument ζ R. In the former case it follows as before that the orbit ends up in E T. In the latter case we follow the arguments in Theorem 3.2 to prove that ( τ, u τ ) cannot end up back in E B, or leave H 4 through the line =. This means that it enters H again at some ζ = ζ Z. However, in this case the incoming part of the orbit is above the part emerging from E B, and therefore the set bounded by {( τ (ζ), u τ (ζ))/ζ < ζ Z } and the graph of from E B to ( τ (ζ Z ), u τ (ζ Z )) is positive invariant. With this, the proof continues as in Theorem 3.2. Theorem 4. states that the orbits go to E T for all τ (, τ ] but it does not state how the orbits behave close to E T. This is given in Proposition 4.4 There exists a τ m > s.t. for τ (, τ m ] any orbit going to E T goes either directly or after a finite number of turns around E T, and for τ (τ m, τ ) the orbit is a stable spiral around E T. Proof To prove this part we use the eigenvalues of the linearization around E T, computed in (4.7). Let τ m = k( T )( ( T )) 2 4c(k ( T ) c). Note that E T is a stable sink for < τ τ m and a stable spiral for τ > τ m. This proves the statement of Proposition 4.4. Having explained the behaviour of orbits close to E T we again turn to existence, this time for τ > τ. As will be seen below, the orbits connecting E B and E T exist for τ > τ too, but to prove this we have to introduce k() G() G( )d T α(, T ) = Figure 2: The functions k, G, and the primitive of G. With this we can now state the main result of this section, G(ϱ;, T )dϱ. (4.2) By the convexity of k, as stated in Assumption (A.), and the definition G(;, ) = k ( )( )+k( ) k(), for any fixed (, ] the function G is decreasing w.r.t. T (, ]. Also, one has G(;, T ) < if (, T ) and G(;, T ) > if ( T, ]. Moreover, α(, ) > > α(, ), (4.3) and α(, ) is decreasing in [, T ]. Observe that α(, T ) does not depend on τ. Figure 2 shows how the functions k(), G() and G(ϱ;, T )dϱ vary with. Theorem 4.2 Let, T (, ], < T and α(, T ) be defined as above. If α(, T ) then for all τ > τ the orbit ( τ, u τ ) reaches E T as ζ. 2